Riemann sums for double integrals practice problems
Estimate a double integral with a midpoint (or corner) Riemann sum on a grid of subrectangles. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
10 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
Estimate \( \displaystyle \iint_R x^{2} + y \, dA \) over \( \displaystyle R = [0, 1] \times [0, 2] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{1} \left(x^{2} + y\right)\, dx\, dy = \frac{8}{3} \]
For comparison, the exact value is 8/3.✓ Proved
Answer \( \frac{5}{4} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R x + y \, dA \) over \( \displaystyle R = [0, 4] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{4} \left(x + y\right)\, dx\, dy = 10 \]
For comparison, the exact value is 10.✓ Proved
Answer \( 10 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R x^{2} + y \, dA \) over \( \displaystyle R = [0, 1] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{1} \left(x^{2} + y\right)\, dx\, dy = \frac{5}{6} \]
For comparison, the exact value is 5/6.✓ Proved
Answer \( \frac{13}{16} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R 8 x y + 4 x + 2 y \, dA \) over \( \displaystyle R = [0, 1] \times [0, 2] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{1} \left(8 x y + 4 x + 2 y\right)\, dx\, dy = 16 \]
For comparison, the exact value is 16.✓ Proved
Answer \( 6 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R 3 x - y^{2} + 5 \, dA \) over \( \displaystyle R = [0, 1] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{1} \left(3 x - y^{2} + 5\right)\, dx\, dy = \frac{37}{6} \]
For comparison, the exact value is 37/6.✓ Proved
Answer \( \frac{45}{8} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R x y + 1 \, dA \) over \( \displaystyle R = [0, 2] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{2} \left(x y + 1\right)\, dx\, dy = 3 \]
For comparison, the exact value is 3.✓ Proved
Answer \( 3 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R x y + 1 \, dA \) over \( \displaystyle R = [0, 2] \times [0, 2] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{2}\int\limits_{0}^{2} \left(x y + 1\right)\, dx\, dy = 8 \]
For comparison, the exact value is 8.✓ Proved
Answer \( 5 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R 8 x y + 4 x + 2 y \, dA \) over \( \displaystyle R = [0, 2] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{2} \left(8 x y + 4 x + 2 y\right)\, dx\, dy = 18 \]
For comparison, the exact value is 18.✓ Proved
Answer \( 18 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R 3 x - y^{2} + 5 \, dA \) over \( \displaystyle R = [0, 1] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 4 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the midpoint points.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{1} \left(3 x - y^{2} + 5\right)\, dx\, dy = \frac{37}{6} \]
For comparison, the exact value is 37/6.✓ Proved
Answer \( \frac{99}{16} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Estimate \( \displaystyle \iint_R x + y \, dA \) over \( \displaystyle R = [0, 2] \times [0, 1] \) with a Riemann sum, \( \displaystyle m = 2 \) subintervals in \( \displaystyle x \) and \( \displaystyle n = 2 \) in \( \displaystyle y \), sampling at the lower left corners.
ΔA times the sum of f at the sample points.✓ Proved
\[ \int\limits_{0}^{1}\int\limits_{0}^{2} \left(x + y\right)\, dx\, dy = 3 \]
For comparison, the exact value is 3.✓ Proved
Answer \( \frac{3}{2} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify